I’m working on a new project with my images, this time an animation called Microspore. I wanted to post the moving version up for you to see, but no matter what I do I can’t get it to look presentable, and the full file at proper resolution is far too big for web streaming. Teh internets are cool, but still w-a-a-a-a-y too slow for serious stuff.

Anyways, here are some stills from the film. You’ll have to imagine that you’re looking through a microscope at little critters drifting slowly past.

Microspore 1

Microspore 2

Microspore 3

Microspore 4

Microspore 5

For your diversion, some eye candy: here are a few of the math structures I’m working with in my artwork at the moment. In these, the images incorporate, as text, parts of the equations that I’ve used to construct the images themselves (you can’t get a lot more recursive than that):

Math Structure 1

Math Structure 2

Math Structure 3


*These particular images don’t have anything at all to do with the Fibonacci Series. I was just trying to come up with a cool headline…


Monty & the Doors

A couple of days back I was reading an article about statistical method in experiments in primate behaviour and the writer mentioned The Monty Hall Problem as a possible source of unintentional introduced error or experimenter bias.

Now The Monty Hall Problem is a fascinating mathematical conundrum, and since I know those kinds of things are always of interest to Cow Readers, I thought those of you who are not familiar with this puzzle might like to exercise your mental muscles on it.

The Monty Hall Problem goes like this:

You are on a game show with your host Monty Hall who is offering you the chance to walk away with the Car of Your Dreams. He shows you three doors, A, B & C.

“The Car of Your Dreams is behind one of these doors,” he says, “The other two doors each conceal a goat. As your Games Master, only I know which door conceals which object. Now, please choose a door to claim your prize!’

You choose your door. You tell Monty “I have chosen Door B!”

“Well done!” he says. “I knew you were a contestant of superior ability! But before we open your door, I’m going to open one of the other doors and show you what’s behind it.” He opens Door C to reveal a goat. “Now that you’ve seen what’s behind Door C,” he says, “I’m going to give you a special opportunity to stick with your chosen door, Door B, or change your choice to the other remaining door, Door A. I’ll give you ten seconds to have a think about it!”

Here’s the question: To win the car, is there any advantage in changing your mind and swapping from your initial choice of Door B to Door A?

Answers on my desk by end of class.

From time to time I get to musing on all manner of things here on The Cow, and today I bring you some thoughts about nature and self-similarity.

This morning at dawn I was lying half-awake listening to the song of a chirpy early-rising Turdus merula, better known as the Common Blackbird. The Blackbird was introduced to Australia, in Melbourne where I am now living, in the 1850s as part of the regrettable We-Wish-It-Was-More-Like-England makeover that the colonists were hell-bent on giving this completely un-Englandlike continent.

The Blackbird’s song is very pretty and very recognizable – listen to the end part of this sample:*

What I realized as I was listening though, was that the little guy† wasn’t just doing the same exact phrase over and over – there were little variations each and every time – just like he was improvising on his little blackbirdy theme. No two riffs were exactly alike.

This got me to speculating about another well-known natural phenomenon in which no two elements are exactly alike, but are very similar in structure and beauty and precision: the snowflake.

Some Snowflakes

And so I began to wonder if the song of the Common Blackbird might in fact be the aural equivalent of the crystalline structure of the snowflake.‡

I don’t really know why I should have made that connection, but there you go. Put it down to my hypnopompic state if you like. But I leave you with this thought: self-similarity is rife in nature. It is embedded everywhere from the mathematics of fractals to the formation of snow crystals and the songs of birds. Its presence is felt in almost every natural process somehow or other. Think about it: it really does not need to be this way. All things considered, the natural world could be completely random. And yet order arises spontaneously everywhere it can.

The reasons for this remain a beautiful mystery.


*This recording by Fred Van Gessel. I pinched it, so for my atonement I urge you to go buy his recording Bird Calls of the Greater Sydney Region from the Australian Museum Shop.

†It was most likely a male territorial song.

‡For a totally absorbing read, go visit this website dedicated to the fastidious, and one must say, obsessive, Wilson Bentley, a man who dedicated his life to the observation and photographic recording of snow crystals.


You know the kind of thing. You get a phone call from your telco saying that they’ve been ‘reviewing’ your plan and they want to help you save money by putting you on a ‘more efficient’ one. Usually this involves re-bundling your mobile phone and your landline services and your internet and your espresso machine in some unfathomable manner. I’ve had two of these calls this week alone and it drives me nuts.

First of all, if it’s true, and you really are spending more money with them than you should be, what the hell kind of logic is there in them telling you that? Could it be that they’re thinking… ‘oooh, we’re making billions of dollars profit but, gosh, we’re so big-hearted that we simply couldn’t sleep at night if we were to rip a few extra cents off our poor customers…’

Yeah, right. Paradoxically, my most recent bill had a whole lot of mysterious SMS ‘status request’ charges that they were quite unwilling to help me ‘be more efficient’ with, even though I didn’t incur them and they couldn’t tell me why they were there.

Following the reasoning that they ‘want to help you save money’ to its logical outcome, let’s say a lot of customers are operating inefficiently, and the company magics those poor suckers’ plans to help them all to spend less. What’s going to happen when it works and the customers put fewer dollars in the telco’s coffers and the company shareholders see a sudden plunge in profits? Right! They’ll put the charges up!!

So it’s plainly illogical that their motive is altruistic. What the hell, then, are they doing.

I stumbled across an article in Choice Magazine recently that may just hold a clue. Here’s the kernel of what it had to say:

The problem is that comparing these bundled packages is next to impossible, even for the pros. Comparison website PhoneChoice says telcos’ pricing structures for bundles are too complex even for quantitative analysis and modelling — there have been several failed attempts to develop systems to compare the offers… Recently, it took a Federal Court judge to adjudicate over which of two mobile phone plans was cheaper.

‘… too complex even for quantitative analysis and modelling…’ Did you get that? We can now make a good fist of modelling the weather on computers, but we can’t simulate competing phone plans.

So here’s my theory: these plans are so complex that not even the people who come up with them can tell what they’re doing! This poses a problem – how, then, do they know which plan is making them more money? Right! They don’t! As a result they have now come to the realization that they must keep swapping their customers from plan to plan in order to have any chance of avoiding diminishing earnings.

They have clevered themselves into confusion.

Let this be a warning to all those with the hubris to challenge complex mathematics.

I have been working with some really interesting generative functions in my artwork and I thought you might like to see some of the results.


(Click on the image and type ‘N’ for Next or ‘P’ for Previous)

The kinds of mathematical systems I’m using for these systems are deeply fascinating. All the images you can see in the above slideshow are closely related, even though they might look substantially different. They seem to resemble complicated organic lifeforms and yet the maths that describes them is remarkably simple.

It works something like this: I outline a basic element, let’s say a small lozenge shape and a circle. Then I tell the maths to do a very simple thing – make a two copies of those shapes in the next generation, displace them in space and rotate them a little. Each subsequent generation executes the same instructions.

This simple set of rules gives rise to a branching structure like you might see in a tree. If I add a few more basic commands (a little random variation in the shapes, some colour change over generations) an astonishing piece of magic happens – the resulting images look organic – even like creatures you might find in the real world. All kinds of phenomena that I don’t specifically code (such as asymmetry and textural effects) appear spontaneously.

I’ve only begun to experiment with these concepts and I fully expect to see some truly wonderful results from this work.

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